# Chapter 26. Capacitance and Dielectrics

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1 Chapter 26 Capacitance and Dielectrics

2 Capacitors Capacitors are devices that store electric charge Examples where capacitors are used: radio receivers filters in power supplies energy-storing devices in electronic flashes

3 Definition of Capacitance The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors C Capacitance is a positive quantity. It is a measure of the ability to store charge The SI unit of capacitance is the farad (F) large, e.g. microfarads (µf) or picofarads (pf) Q V

4 Makeup of a Capacitor A capacitor consists of two conductors These are called plates When the plates are charged, they have equal and opposite charges A potential difference exists between the plates due to this charge

5 Quick Quiz 26.1 A capacitor stores charge Q at a potential difference ΔV. If the voltage applied by a battery to the capacitor is doubled to 2ΔV: (a) the capacitance falls to half its initial value and the charge remains the same (b) the capacitance and the charge both fall to half their initial values (c) the capacitance and the charge both double (d) the capacitance remains the same and the charge doubles

6 Quick Quiz 26.1 Answer: (d). The capacitance is a property of the physical system and does not vary with applied voltage. According to C=Q/ V, if the voltage is doubled, the charge is doubled.

7 Demo Ea9: Parallel plate capacitor and plate separation Voltage proportional to distance between plates V=Ed Voltage decreases on inserting dielectric between plates V = Q/κC 0

8 Parallel Plate Capacitor Each plate is connected to a terminal of the battery Suppose initially uncharged Battery then establishes an electric field in the connecting wires

9 Parallel Plate Capacitor Consider, first, the negative terminal: Field applies a force on electrons in the wire Force causes the electrons to move onto the negative plate Continues until equilibrium is achieved i.e. the plate, wire and terminal are all at the same potential There is now no field present in the wire Hence the movement of electrons ceases The plate is now negatively charged

10 Parallel Plate Capacitor Now consider the positive terminal A similar process occurs at the other plate, with electrons moving away from the plate and leaving it positively charged In its final configuration, the potential difference across the capacitor plates is the same as that between the terminals of the battery

11 Capacitance Parallel Plates Charge density σ = Q/A Electric field E = / 0 (for conductor) Uniform between plates, zero elsewhere C Q V Q Ed Q Q 0 A d 0 A i.e. proportional to the area of plates and inversely proportional to the distance between them d

12 Parallel Plate Assumptions Assumption that electric field is uniform is valid in the central region, but not at the ends of the plates If separation between plates is small compared with their length, effect of non-uniform field can be ignored

13 Quick Quiz 26.2 Many computer keyboard buttons are constructed of capacitors, as shown in the figure below. When a key is pushed down, the soft insulator between the movable plate and the fixed plate is compressed. When the key is pressed, the capacitance (a) increases (b) decreases (c) changes in a way that we cannot determine because the complicated electric circuit connected to the keyboard button may cause a change in ΔV.

14 Quick Quiz 26.2 Answer: (a). When the key is pressed, the plate separation is decreased and the capacitance increases (since C= 0 A/d). Capacitance depends only on how a capacitor is constructed and not on the external circuit.

15 Capacitance of Isolated Sphere Assume a spherical charged conductor Assume V = 0 at infinity. Then C Q V Q k e Q / R R k e 4 0 R independent of charge and potential difference

16 Spherical Capacitor (ex 26.3) V b a E r dr k e Q b dr k Q 1 r 2 e r a a b k e Q 1 b 1 a C Q V 1 1 k e 1 a b

17 Cylindrical Capacitor (ex. 26.2) Q= l E = 2k e / r From Gauss s Law (exercise ) V C Q V b a E r dr b dr 2 k e 2 k e ln b r a l 2 k e ln b a a

18 Geometry of Some Capacitors

19 Circuit Symbols A circuit diagram is a simplified representation of an actual circuit Circuit symbols are used to represent the various elements Lines are used to represent wires The battery s positive terminal is indicated by the longer line

20 ECM05ANA: Charging & Discharging a Capacitor

21 Capacitors in Parallel When capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged

22 Capacitors in Parallel, 2 The flow of charges ceases when the voltage across the capacitors equals that of the battery Maximum charge Total charge is sum of the charges Q total = Q 1 + Q 2 Potential difference across the capacitors is the same, equal to voltage of battery (V=V 1 =V 2 ) Hence Q total /V = Q 1 /V 1 + Q 2 /V 2, so C eq = C 1 + C 2

23 Capacitors in Parallel, 3 Capacitors can be replaced with one capacitor with a capacitance of C eq The equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors

24 Capacitors in Series When a battery is connected to the circuit, electrons are transferred from the left plate of C 1 to the right plate of C 2 through the battery

25 Capacitors in Series, 2 As negative charge accumulates on right plate of C 2, an equivalent amount of negative charge is removed from left plate of C 2, leaving it with excess positive charge All the right plates gain charges of Q, all left plates have charges of +Q

26 Capacitors in Series, 3 The potential differences add up to the battery voltage Q Q 1 Q 2 V V 1 V 2 V Q V 1 Q 1 V 2 Q 2 1 C 1 C 1 1 C 2

27 Capacitors in Combination When two or more capacitors are connected in parallel, the potential differences across them are the same Charge on each capacitor proportional to its capacitance Capacitors add directly to give the equivalent capacitance When two or more capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same Capacitances add as reciprocals Equivalent capacitance always less than smallest individual capacitor

28 Equivalent Capacitance (exercise) The 1.0-µF and 3.0-µF capacitors are in parallel as are the 6.0-µF and 2.0-µF capacitors These parallel combinations are in series with the capacitors next to them The series combinations are in parallel and the final equivalent capacitance can be found

29 Quick Quiz 26.3 Two capacitors are identical. They can be connected in series or in parallel. If you want the smallest equivalent capacitance for the combination, you should connect them in (a) series (b) parallel (c) Either combination has the same capacitance.

30 Quick Quiz 26.3 Answer: (a). When connecting capacitors in series, the inverses of the capacitances add (1/C = 1/C 1 + 1/C 2 ), resulting in a smaller overall equivalent capacitance.

31 Quick Quiz 26.4 Consider two identical capacitors. Each capacitor is charged to a voltage of 10 V. If you want the largest combined potential difference across the combination, you should connect them in (a) series (b) parallel (c) Either combination has the same potential difference.

32 Quick Quiz 26.4 Answer: (a). When capacitors are connected in series, the voltages add, for a total of 20 V in this case. If they are combined in parallel, the voltage across the combination is still 10 V.

33 Exercise Connect plates of capacitor to battery. What happens to the charge when the connecting wires to the battery are removed? Nothing! Charge remains on the plates. What happens if the wires are now connected to one another? Charges move along wires and plates until the entire conductor is at a single potential and the capacitor is discharged.

34 Demo Eb14 Energy Storage in Capacitor Charge up capacitor using DC power supply. Disconnect and attach leads to electric motor. Rotation of motor enables work to be done.

35 ECA05AN2: Energy storage in a capacitor

36 Energy in a Capacitor Overview Before switch is closed, energy is stored as chemical energy in battery When switch is closed, energy is then transformed from chemical to electric potential energy

37 Energy in a Capacitor Overview, cont Electric potential energy related to separation of positive and negative charges on plates Thus, a capacitor is a device that stores energy as well as charge

38 Energy Stored in a Capacitor Assume capacitor is being charged and, at some point, has a charge q on it and a potential difference V The work then needed to transfer a charge dq from one plate to the other is d W V d q q d q C The total work required is W Q q d q Q C 0 C 2 2

39 Energy, cont The work done in charging the capacitor appears as electric potential energy U: (remember C = Q/ V) Applies in any geometry 2 U Q 1 Q V 1 C ( V ) 2C 2 2 Energy stored increases as charge increases and as potential difference increases In practice, there is a maximum voltage before discharge occurs between the plates 2

40 Energy, final Energy is stored in the electric field For parallel-plate capacitor, the energy can be expressed in terms of the field: U = ½ C V 2 =½ (ε o A/d)(Ed) 2 = ½ (ε o Ad)E 2 Can also be expressed as the energy density (energy per unit volume [Ad]) u E = U/[Ad] = ½ ε o E 2

41 Quick Quiz 26.5 You have three capacitors and a battery. In which of the following combinations of the three capacitors will the maximum possible energy be stored when the combination is attached to the battery? (a) series (b) parallel (c) Both combinations will store the same amount of energy.

42 Quick Quiz 26.5 Answer: (b). For a given voltage, the energy stored in a capacitor is proportional to C: U = C(ΔV) 2 /2. Thus, you want to maximize the equivalent capacitance and the potential difference cross it. You do this by connecting the three capacitors in parallel, so that the capacitances add, and each capacitor has the same potential difference, ΔV, across it.

43 Quick Quiz 26.6 You charge a parallel-plate capacitor, remove it from the battery, and prevent the wires connected to the plates from touching each other. When you pull the plates apart to a larger separation, do the following quantities increase, decrease, or stay the same? (a) C; (b) Q; (c) E between the plates; (d) ΔV ; (e) energy stored in the capacitor.

44 Quick Quiz 26.6 Answers: (a) C decreases (C= 0 A/d). (b) Q stays the same because there is no place for the charge to flow. (c) E remains constant (E= /2 0 ). (d) ΔV increases because ΔV = Q/C, Q is constant (part b), and C decreases (part a). (e) The energy stored in the capacitor is proportional to both Q and ΔV 2 and thus increases. The additional energy comes from the work you do in pulling the two plates apart.

45 Quick Quiz 26.7 Repeat Quick Quiz 26.6, but this time answer the questions for the situation in which the battery remains connected to the capacitor while you pull the plates apart. Do this in your own time.

46 Quick Quiz 26.7 Answers: (a) C decreases (C= 0 A/d). (b) Q decreases. The battery supplies a constant potential difference ΔV; thus, charge must flow out of the capacitor if C = Q /ΔV is to decrease. (c) E decreases because the charge density on the plates decreases. (d) ΔV remains constant because of the presence of the battery. (e) The energy stored in the capacitor decreases (U = C(ΔV) 2 /2).

47 Some Uses of Capacitors Defibrillators When fibrillation occurs, the heart produces a rapid, irregular pattern of beats A fast discharge of electrical energy through the heart can return the organ to its normal beat pattern In general, capacitors act as energy reservoirs that can be slowly charged and then discharged quickly to provide large amounts of energy in a short pulse

48 Demo Eb4 Energy storage in a capacitor Light bulb placed in series with a capacitor. Currents generated in charging and discharging the capacitor. Note that light bulb does not have constant resistance as the temperature changes.

49 Capacitors with Dielectrics A dielectric is a nonconducting material that, when placed between the plates of a capacitor, increases the capacitance Dielectrics include rubber, plastic, and waxed paper For a parallel-plate capacitor C = κc o = κε o (A/d) The capacitance is multiplied by the factor κ when the dielectric completely fills the region between the plates

50 Dielectrics, cont In theory, d could be made very small to create a very large capacitance In practice, there is a limit to how small: d is limited by the electric discharge that could occur though the dielectric medium separating the plates For a given d, the maximum voltage that can be applied to a capacitor without causing a discharge depends on the dielectric strength of the material

51 Dielectrics, final Dielectrics provide the following advantages: Increase in capacitance Increase the maximum operating voltage Possible mechanical support between the plates Allows plates to be close together without touching This decreases d and increases C

52

53 Dielectrics An Atomic View (not examinable) The molecules that make up the dielectric are modeled as dipoles The molecules are randomly oriented in the absence of an electric field

54 Dielectrics Atomic View, 2 An external electric field is applied This produces a torque on the molecules The molecules partially align with the electric field

55 Induced Charge and Field The electric field due to the plates is directed to the right and it polarizes the dielectric The net effect on the dielectric is an induced surface charge that results in an induced electric field If the dielectric were replaced with a conductor, the net field between the plates would be zero

56 Quick Quiz 26.9 A fully charged parallel-plate capacitor remains connected to a battery while you slide a dielectric between the plates. Do the following quantities increase, decrease, or stay the same? (a) C; (b) Q; (c) E between the plates; (d) ΔV.

57 Quick Quiz 26.9 Answers: (a) C increases (C= C 0 ). (b) Q increases. Because the battery maintains a constant ΔV, Q must increase if C increases (Q=C V). (c) E between the plates remains constant because ΔV = Ed and neither ΔV nor d changes. The electric field due to the charges on the plates increases because more charge has flowed onto the plates. However, the induced surface charges on the dielectric create a field that opposes the increase in the field caused by the greater number of charges on the plates. (d) The battery maintains a constant ΔV.

58 End of Chapter

59 Dielectrics Atomic View, 3 Degree of alignment of the molecules with the field depends on temperature and the magnitude of the field In general, the alignment increases with decreasing temperature the alignment increases with increasing field strength

60 Dielectrics Atomic View, 4 If the molecules of the dielectric are nonpolar molecules, the electric field produces some charge separation This produces an induced dipole moment The effect is then the same as if the molecules were polar

61 Dielectrics Atomic View, final An external field can polarize the dielectric whether the molecules are polar or nonpolar The charged edges of the dielectric act as a second pair of plates producing an induced electric field in the direction opposite the original electric field

62 Quick Quiz 26.8 If you have ever tried to hang a picture or a mirror, you know it can be difficult to locate a wooden stud in which to anchor your nail or screw. A carpenter s stud-finder is basically a capacitor with its plates arranged side by side instead of facing one another, as shown in the figure below. When the device is moved over a stud, the capacitance will: (a) increase (b) decrease

63 Quick Quiz 26.8 Answer: (a). The dielectric constant of wood (and of all other insulating materials, for that matter) is greater than 1; therefore, the capacitance increases (C= C 0 ). This increase is sensed by the stud-finder's special circuitry, which causes an indicator on the device to light up.

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